nonlinear dynamics – phenomena and applications ali h. nayfeh department of engineering science...
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Nonlinear Dynamics – Phenomena and Applications
Ali H. NayfehAli H. Nayfeh
Department of Engineering Science and MechanicsDepartment of Engineering Science and Mechanics
Virginia TechVirginia Tech
Lyapunov Lecture
The 2005 ASME International Design Engineering Technical Conferences
24-28 September 2005
Lyapunov Lecture 2005
Outline
Parametric Instability in Ships The Parametric Instability in Ships The Saturation PhenomenonSaturation Phenomenon
Exploitation of the Saturation Phenomenon Exploitation of the Saturation Phenomenon for Vibration Controlfor Vibration Control
Transfer of Energy from High-to-Low Transfer of Energy from High-to-Low Frequency ModesFrequency Modes
Crane-Sway ControlCrane-Sway Control From theory to laboratory to fieldFrom theory to laboratory to field Ship-mounted cranesShip-mounted cranes Container cranesContainer cranes
Concluding RemarksConcluding Remarks
Lyapunov Lecture 2005
A recent accident attributed to A recent accident attributed to parametric instabilityparametric instability A C11 class container ship suffered a A C11 class container ship suffered a
parametric instability of over 35 degrees in parametric instability of over 35 degrees in rollroll
Many containers were thrown overboardMany containers were thrown overboard Shipper sued ship owner for negligent Shipper sued ship owner for negligent
operationoperation Case was settled out of court Case was settled out of court
Parametric Instability in Ships
Lyapunov Lecture 2005
L : 223.5 cmB : 29.2 cm D : 19.1 cmW: 30.5 kg without ballastW: 54.5 kg with ballast
•Roll frequency : 0.32 Hz
•Wave frequency: 0.60 Hz
Parametric Instability in a Tanker Model
Only pitch and heave are directly excited
Virginia Tech 1991I. Oh
Lyapunov Lecture 2005
Laboratory Results on a Tanker ModelVirginia Tech 1991
Lyapunov Lecture 2005
Autoparametric Instability in Ships
In 1863, Froude remarked in the In 1863, Froude remarked in the Transactions of the British Institute of Transactions of the British Institute of Naval Architects thatNaval Architects that
a ship whose frequency in heave (pitch) is a ship whose frequency in heave (pitch) is twice its frequency in roll has undesirable twice its frequency in roll has undesirable sea keeping characteristicssea keeping characteristics
Lyapunov Lecture 2005
Destroyer Model in a Regular Head Wave
• Model: US Navy Destroyer Hull # 4794• Bare Hull Model Roll freq. : 1.40 Hz Pitch freq. : 1.65 Hz Heave freq.: 1.45 Hz
• Model with Ballast Roll freq. : 0.495 Hz Pitch freq. : 0.910 Hz Heave freq.: 1.260 Hz
• Wave freq. : 0.90 Hz
Only pitch and heave are directly excited
Virginia Tech 1991I. Oh
Lyapunov Lecture 2005
A Possible Explanation of Froude’s Remark
Roll and pitch motions are uncoupled linearlyRoll and pitch motions are uncoupled linearly
22 0r r
2 22 cos( )p p F t
• They are coupled nonlinearly- A paradigm
Larry Marshal & Dean Mook
p2 andp r
Lyapunov Lecture 2005
Perturbation Solution
212
1cos ta
)(cos 2 tb
• Pitch response:
• Method of Multiple Scales or Method of Averaging Perturbation Methods with Maple: http://www.esm.vt.edu/~anayfeh/
Perturbation Methods with Mathematica: http://www.esm.vt.edu/~anayfeh/
• Roll response:
Lyapunov Lecture 2005
Equilibrium Solutions
• Linear response
0a2 p
Fb
and
• Nonlinear response
( , , , , , , )r p r pa f F
2 2 2( 2 )r r r
r
b
Independent of
Excitation Amp. F
Lyapunov Lecture 2005
Response Amplitudes The Saturation Phenomenon
LinearResponse
Response after Saturation
b
a
a
Pitch Amplitude
Roll Amplitude
Wave Height
b Pitch Amplitude
Lyapunov Lecture 2005
Exploitation of the Saturation Phenomenon for Vibration Control
The ship pitch is replaced with a mode of the plantThe ship pitch is replaced with a mode of the plant The ship roll is replaced with an electronic circuitThe ship roll is replaced with an electronic circuit The mode of the plant is coupled quadratically to The mode of the plant is coupled quadratically to
the electronic circuitthe electronic circuit The coupling is effected by an actuator and a The coupling is effected by an actuator and a
sensorsensor ActuatorActuator
Piezoceramic or magnetostrictive or electrostrictive Piezoceramic or magnetostrictive or electrostrictive materialmaterial
SensorSensor Strain gauge or accelerometerStrain gauge or accelerometer
Shafic Oueini, Jon Pratt, and Osama Ashour
Lyapunov Lecture 2005
Absorber
• Plant model22 cos( )p p cu u u F t F
p • Equations of controller and control signal
22 c cv v v u v 21
and2c cF v
Lyapunov Lecture 2005
Perturbation Solution
1 2
1cos
2v a t
2cos( )u b t
Lyapunov Lecture 2005
Equilibrium Solutions
• Linear response
0a2 p
Fb
and
• Nonlinear response
( , , , , , , )c p c pa f F
2 2 2( 2 )c c c
c
b
Independent of
Excitation Amp. F
Lyapunov Lecture 2005
Bifurcation Analysis
a,b
FLinear
ResponseResponse after Saturation
(Region of Control)
b
a
a
2 2 2( 2 )c c c
c
b
Lyapunov Lecture 2005
Optimal Absorber Frequency
1
2c
c b
ControllerDamping
FeedbackGain
0bPlant Response
Amplitude
2 2 2( 2 )c c c
c
b
Plant Amplitude
Lyapunov Lecture 2005
Experiments
Beams and PlatesBeams and Plates ActuatorsActuators
Piezoceramic patchesPiezoceramic patches Magnetostrictive unbiased Terfenol-DMagnetostrictive unbiased Terfenol-D
SensorsSensors Strain gaugeStrain gauge AccelerometerAccelerometer
ImplementationImplementation AnalogAnalog DigitalDigital
Lyapunov Lecture 2005
Sensor and ActuatorConfiguration
PiezoceramicActuators
Strain Gauge
ShakerFixture
Lyapunov Lecture 2005
Single-Mode Controlz
0 100 200 300
Time (sec)
-0 .50
0.00
0.50
Str
ain
(V
)
mgmgF 9.88.5
Lyapunov Lecture 2005
Amplitude-Response Curve
z
0.00 20.00 40.00 60.00 80.00
Forcing Am plitude (m g)
0.00
10.00
Str
ain
(V
)
Open-Loop
Closed-Loop
Lyapunov Lecture 2005
Frequency-Response Curve
F = 30mg
10.00 10.40 10.80 11.20 11.60
Forcing Frequency (Hz)
0.00
2.00
4.00
6.00
Str
ain
(V
)
Open-Loop
Closed-Loop
Lyapunov Lecture 2005
Control of Plates
A schematic of a cantilever plate with a PZT actuator
Lyapunov Lecture 2005
17.2 17.6 18 18.4 18.8
F requency (Hz)
-30
-20
-10
0
10
Str
ain
(d
B)
0 4 8 12 16 20
Inpu t Shaker Acce le ra tion (m g)
0
1
2
3
4
5
Str
ain
(m
V)
Frequency -response curves Force-response curves
Response Curves
Lyapunov Lecture 2005
Zero-to-One Internal Resonance
Natural frequencies: 0.65, 5.65, 16.19, 31.91 HzNatural frequencies: 0.65, 5.65, 16.19, 31.91 Hz
f = 16.23 Hz
T. Anderson, B. Balachandran, Samir Nayfeh, P. Popovic, M. Tabaddor, K. Oh, H. Arafat, and P. Malatkar
Lyapunov Lecture 2005
Natural frequencies: 0.70, 5.89, 16.75, 33.10, 54.40 HzNatural frequencies: 0.70, 5.89, 16.75, 33.10, 54.40 Hz
f = 32.20 Hz
Zero-to-One Internal ResonanceExternal Excitation
Lyapunov Lecture 2005
Zero-to-One Internal ResonanceParametric Excitation
Natural frequencies: 0.65, 5.65, 16.19, 31.91 HzNatural frequencies: 0.65, 5.65, 16.19, 31.91 Hz
f = 32.289 Hz
Lyapunov Lecture 2005
Simultaneous One-to-Oneand Zero-t-one Resonances
Natural Frequencies:Natural Frequencies:1.303, 9.049, 25.564,1.303, 9.049, 25.564,50.213, 83.105 Hz50.213, 83.105 Hz
• Excitation frequency:
83.5 Hz near the fifth
natural frequency
• Large response at
1.3 Hz : first-mode
frequency
Lyapunov Lecture 2005
One-to-One Internal ResonanceWhirling Motion
Natural Frequencies:Natural Frequencies:1.303, 9.049, 25.564,1.303, 9.049, 25.564,50.213, 83.105 Hz50.213, 83.105 Hz
• Excitation frequency:
84.9 Hz near the fifth
natural frequency
Lyapunov Lecture 2005
Natural Frequencies:Natural Frequencies:1.303, 9.049, 25.564,1.303, 9.049, 25.564,50.213, 83.105 Hz50.213, 83.105 Hz
• Excitation frequency:
84.5 Hz near the fifth
natural frequency
One-to-One Internal ResonanceWhirling Motion
Note the reverse in the direction of whirl
Lyapunov Lecture 2005
Natural Frequencies:Natural Frequencies:1.303, 9.049, 25.564,1.303, 9.049, 25.564,50.213, 83.105 Hz50.213, 83.105 Hz
• Excitation frequency:
84.98 Hz near the fifth
natural frequency
• Large response at 1.3 Hz :
first-mode frequency
Simultaneous One-to-Oneand Zero-t-one Resonances
Lyapunov Lecture 2005
Natural Frequencies: 1.303, 9.049, 25.564, 50.213, 83.105 Hz
f = 83.5
Simultaneous One-to-Oneand Zero-t-one Resonances
Lyapunov Lecture 2005
A Paradigm for Zero-to-One Resonance
21
tfuuuuuu
uuuuuu
cos2
2
2214
323222
222
2212
311111
211
Samir Nayfeh
Lyapunov Lecture 2005
Nondimensionalization
We introduce a small parameterWe introduce a small parameter
We introduce nondimensional quantitiesWe introduce nondimensional quantities
Nondimensional equationsNondimensional equations
21 /
22221112 ,,,/ ucuucutt
)cos(2
)4(2
2214
3232222
2212
311
2111
21
tfuuuuuu
uuuuuu
Lyapunov Lecture 2005
Variation of Parameters
We letWe let
Detuning from resonanceDetuning from resonance
)(
)](sin[)(
)](cos[)(
2
2
11
tt
tttau
tttau
vu
12
Lyapunov Lecture 2005
Variational Equations
)cossin2
coscoscos(cos
)cossin2
coscoscos(sin
2
214
333
2
214
333
tfa
auaaa
tfa
auaaa
)cos42( 2212
3111111
11
auuvuv
vu
Lyapunov Lecture 2005
Averaged Equations--Modulation Equations
)cos28
321
21
(
)sin21
(
23
214
2
af
au
faa
)21
42( 212
3111111
11
auuvuv
vu
Lyapunov Lecture 2005
Equilibrium Solutionsor Fixed Points
021
4
02
123111
1
auuu
v
0cos43
0sin22
3214
2
af
au
fa
Lyapunov Lecture 2005
Two Possible Fixed Points
FirstFirst
Second mode oscillates around an undeflected first modeSecond mode oscillates around an undeflected first mode
SecondSecond
Second mode oscillates around a statically deflected first modeSecond mode oscillates around a statically deflected first mode
01 u 222
22
3 443
a
fa
222
2214
23
1
22
1
443
82
a
fua
au
Lyapunov Lecture 2005
Frequency-Response Curves
3,1
2,1
43
21
Lyapunov Lecture 2005
Ship-Mounted CraneUncontrolled ResponseUncontrolled Response
Animation is faster Animation is faster than real timethan real time
2° Roll at 2° Roll at nn
1° Pitch at 1° Pitch at nn
1 ft Heave at 21 ft Heave at 2nn
Ziyad Masoud
Lyapunov Lecture 2005
Control Strategy
Control boom luff and slew angles, which Control boom luff and slew angles, which are already actuatedare already actuated
Time-delayed position feedback of the Time-delayed position feedback of the load cable angles. For the planar motion,load cable angles. For the planar motion,
delay time theis and gain, a is
position, reference some are and where
)](sin[)()(
)](sin[)()(
00
0
0
k
yx
tkltyty
tkltxtx
outp
inp
Lyapunov Lecture 2005
Damping
Lyapunov Lecture 2005
Controlled Response
Animation is faster Animation is faster than real timethan real time
2° Roll at 2° Roll at nn
1° Pitch at 1° Pitch at nn
1 ft Heave at 21 ft Heave at 2nn
Lyapunov Lecture 2005
Controlled vs. Uncontrolled Response (Fixed Crane Orientation)
Lyapunov Lecture 2005
Controlled vs. Uncontrolled Response (Fixed Crane Orientation)
Lyapunov Lecture 2005
Controlled Response
Slew OperationSlew OperationAnimation is faster Animation is faster
than real timethan real time2° Roll at 2° Roll at nn
1° Pitch at 1° Pitch at nn
1 ft Heave at 21 ft Heave at 2nn
Lyapunov Lecture 2005
Controlled vs. Uncontrolled Response (Slewing Crane)
Lyapunov Lecture 2005
Controlled vs. Uncontrolled Response (Slewing Crane)
Lyapunov Lecture 2005
Performance of Controllerin Presence of Initial Disturbance
Animation is faster Animation is faster than real timethan real time
2° Roll at 2° Roll at nn
1° Pitch at 1° Pitch at nn
1 ft Heave at 21 ft Heave at 2nn
Lyapunov Lecture 2005
Experimental Demonstration
A 3 DOFA 3 DOF ship-motionship-motion simulator simulator platform is built:platform is built:
It has the capability of performing general pitch, roll, and heave motions
A 1/24 scale model of the T-ACS (NSWC) crane is mounted on the platform
A PC is used to apply the controller and drive the crane
A 3 DOFA 3 DOF ship-motionship-motion simulator simulator platform is built:platform is built:
It has the capability of performing general pitch, roll, and heave motions
A 1/24 scale model of the T-ACS (NSWC) crane is mounted on the platform
A PC is used to apply the controller and drive the crane
Ziyad Masoud and Ryan Henry
Lyapunov Lecture 2005
Uncontrolled Response
1° Roll at 1° Roll at nn
0.5° Pitch at 0.5° Pitch at nn
0.5 in Heave at 20.5 in Heave at 2nn
Lyapunov Lecture 2005
Controlled Response
2° Roll at 2° Roll at nn
1° Pitch at 1° Pitch at nn
0.5 in Heave at 20.5 in Heave at 2nn
Lyapunov Lecture 2005
Controlled Response Slewing Crane
2° Roll at 2° Roll at nn
1° Pitch at 1° Pitch at nn
0.5 in Heave at 20.5 in Heave at 2nn
Lyapunov Lecture 2005
Performance of Controller(in Presence of Initial Conditions)
Lyapunov Lecture 2005
Container Cranes
Lyapunov Lecture 2005
65-Ton Container CraneCommanded Cargo
TrajectoryCommanded Cargo
Trajectory
Lyapunov Lecture 2005
65-Ton Container Crane
Uncontrolled SimulationUncontrolled Simulation
The animation is The animation is twice as fast as the twice as fast as the actual speedactual speed
Lyapunov Lecture 2005
65-Ton Container Crane
Controlled SimulationControlled SimulationThe animation is The animation is
twice as fast as the twice as fast as the actual speedactual speed
Lyapunov Lecture 2005
65-Ton Container Crane
Full-Scale Simulation ResultsFull-Scale Simulation Results
Lyapunov Lecture 2005
Experimental Validation on IHI 1/10th Scale Model
Load PathLoad Path
Lyapunov Lecture 2005
IHI Model
Ziyad Masoud and Nader Nayfeh
Lyapunov Lecture 2005
Experimental ResultsIHI Model
Lyapunov Lecture 2005
Manual Mode - UncontrolledIHI Model
HalfSpeed
Lyapunov Lecture 2005
Manual Mode - ControlledIHI Model
Lyapunov Lecture 2005
Experimental ValidationVirginia Tech Model
Ziyad Masoud and Muhammad Daqaq
Lyapunov Lecture 2005
Manual Mode - Uncontrolled Virginia Tech Model
HalfSpeed
Lyapunov Lecture 2005
Manual Mode - Controlled Virginia Tech Model
Lyapunov Lecture 2005
Pendulation ControllerController can suppress cargo sway inController can suppress cargo sway in
Commercial cranesCommercial cranesMilitary cranesMilitary cranes
Effectiveness of the Controller has been Effectiveness of the Controller has been demonstrated using computer models of demonstrated using computer models of Ship-mounted boom cranesShip-mounted boom cranesLand-based rotary cranesLand-based rotary cranes65-ton container crane65-ton container craneTelescopic craneTelescopic crane
Controller has been validated experimentally on Controller has been validated experimentally on scaled models ofscaled models ofShip-mounted boom craneShip-mounted boom craneLand-based rotary craneLand-based rotary craneContainer crane in an industrial settingContainer crane in an industrial settingFull-scale container craneFull-scale container crane
Lyapunov Lecture 2005
Concluding Remarks
Nonlinearities pose challenges and Nonlinearities pose challenges and opportunities opportunities
ChallengesChallenges Design systems that overcome the adverse Design systems that overcome the adverse
effects of nonlinearitieseffects of nonlinearities Develop passive and active control strategies Develop passive and active control strategies
to expand the design envelopeto expand the design envelope
OpportunitiesOpportunities Exploit nonlinearities for designExploit nonlinearities for design
Is nonlinear thinking in order
?Lyapunov Lecture 2005
Lyapunov Lecture 2005
Controller
Nonlinear delay feedback controlNonlinear delay feedback control
PID Plant
GainCalculatorController
+
+
+
-
T
k
Lyapunov Lecture 2005
Typical Terfenol-D Strut
Prestress housing
Prestress spring
Solenoid
Magnet
Terfenol-D
Lyapunov Lecture 2005
Terfenol-DConstitutive Law
Field (H)
Bias line
Linear operation
Nonlinearoperation
Nonlinearoperation
Lyapunov Lecture 2005
SetupShaker Excitation
Terfenol-DActuator
Shaker
Accelerometer
Shafic Oueini & Jon Pratt
Lyapunov Lecture 2005
Single-Mode Control
z
0 10 20 30 40
Time (sec)
-0 .50
0.00
0.50
Acc
eler
atio
n (
g)
Lyapunov Lecture 2005
Required Luff Rate
Using the motions of the Bob Hope obtained with the integrated Stabilization System, we calculated the crane luff rates demanded by the controller and compared them with the rates supplied by MacGregor
Jib angular rate vs maximum controlled rate
0.00
2.00
4.00
6.00
8.00
10.00
12.00
0.00 10.00 20.00 30.00 40.00 50.00 60.00 70.00 80.00 90.00
jib angle (deg)
jib r
ate
(deg
/s)
max crane rate
max control commanded rate
Lyapunov Lecture 2005
Summary
Anti-Roll TanksAnti-Roll Tanks Demonstrated the benefits of active anti-roll tanks in regular and irregular seas Demonstrated the benefits of active anti-roll tanks in regular and irregular seas
((for all headingsfor all headings)) A thirty-fold roll reduction with a tank mass= 0.6 % ship mass for all headings in SS5A thirty-fold roll reduction with a tank mass= 0.6 % ship mass for all headings in SS5 Less than 0.5Less than 0.5° roll° roll
Fender and Mooring SubsystemFender and Mooring Subsystem Developed a control strategy to maintain a skin-to-skin configuration between Developed a control strategy to maintain a skin-to-skin configuration between
two shipstwo ships
Prevents metal-on-metal contact between two shipsPrevents metal-on-metal contact between two ships
Minimizes the motions of the Bob Hope and the ArgonautMinimizes the motions of the Bob Hope and the Argonaut
Limits the motion of the Argonaut relative to the Bob HopeLimits the motion of the Argonaut relative to the Bob Hope
Reduces the demand on cranesReduces the demand on cranes
Enables operations in SS4 & SS5Enables operations in SS4 & SS5
Decreases the transfer timeDecreases the transfer time
Lyapunov Lecture 2005
Effectiveness of Mooring System
0.00
2.00
4.00
6.00
8.00
10.00
12.00
10.00 20.00 30.00 40.00 50.00 60.00 70.00 80.00
Commanded Crane Angle (degrees)
Ra
te (
de
g/s
)
max crane speed
SS5 controller requirements - 20° following off stern of Bob Hope
SS4 controller requirements - 15° off head seas
SS5 controller requirements - 15° off head seas
Lyapunov Lecture 2005
Controller
Nonlinear delay feedback controlNonlinear delay feedback control Nonlinear delay feedback controlNonlinear delay feedback control
PID Plant
GainCalculatorController
+
+
+
-
T
k
Lyapunov Lecture 2005
The Control Unit
Trolley
Hoist 1
Hoist 2
Sway
Joystick - Trolley
Joystick - Hoist
Quadrature Encoder Input
ADC
Trolley Motor
Hoist 2 Motor
Hoist 1 MotorDACControl Unit
Lyapunov Lecture 2005
Controller Circuit Piezoceramic Actuator
System
1K 2Ks1
s2
DK
2vu
v
vvu
Lyapunov Lecture 2005
Nonresonance InteractionZero-to-One Internal Resonance
Natural frequencies: 0.65, 5.65, 16.19, 31.91 HzNatural frequencies: 0.65, 5.65, 16.19, 31.91 Hz
f = 16.25 Hz
Lyapunov Lecture 2005
Comparison between Responses of Beam and Hubble Telescope
Lyapunov Lecture 2005
IHI Scale Model Profile